Testing equality of autocovariance operators for functional time series

Abstract

We consider strictly stationary stochastic processes of Hilbert space-valued random variables and focus on tests of the equality of the lag-zero autocovariance operators of several independent functional time series. A moving block bootstrap-based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomes some common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a fully functional test statistic based on the Hilbert-Schmidt distance of the corresponding sample lag-zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the moving block bootstrap procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested moving block bootstrap-based testing procedure is illustrated through simulations and an application to a real-life dataset is discussed.

Testing Relevant Hypotheses in Functional Time Series via Self-Normalization

“…Pilavakis et al . () proposed a test for the equality of the lag 0 autocovariance operators of K functional time series, and Sharipov and Wendler () considered bootstrap‐based statistical inference for covariance operators of functional time series. Change point analysis for covariance operators was developed by Aston and Kirch () and Stoehr et al .…”

Tests for conditional heteroscedasticity of functional data

“…To achieve this one could implement a covariance block bootstrap approach as described in Zhang () and Pilavakis et al . (), but for the sake of computational speed, and due to its satisfactory performance, we instead propose to estimate the limiting distribution directly. This could be done by estimating the eigenvalues of the kernel integral operator with kernel via estimates of the kernel, or alternatively using a Welch–Satterthwaite style approximation, the later of which we pursue; see for example, Zhang ().…”

Detecting relevant differences in the covariance operators of functional time series -- a sup-norm approach

“…Boente et al (2018) provide a theoretical framework which clarifies the ability of the test to detect local alternatives. Pilavakis et al (2020) develop a fully functional test for the equality of auto-covariance operators of temporally dependent time series, which is based on a moving block bootstrap. For independent data the K-sample problem has also been considered by Guo et al (2016) who propose to estimate the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function.…”